# Tour:Subgroup containment relation equals divisibility relation on generators

From Groupprops

**This article adapts material from the main article:** subgroup containment relation in the group of integers equals divisibility relation on generators

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Every nontrivial subgroup of the group of integers is cyclic on its smallest element|UP: Introduction four (beginners)|NEXT: No proper nontrivial subgroup implies cyclic of prime order

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WHAT YOU NEED TO DO:

- Read and understand the main statement, as well as the corollaries.
- Convince yourself of the truth of the main statement, as well as of how the corollaries follow from it.

## Statement

Let denote the Group of integers (?) under addition. Suppose are subgroups of . Suppose and . Then, if and only if , i.e., is a multiple of .

Note that, because every nontrivial subgroup of the group of integers is cyclic on its smallest element, the subgroups and *can* be written in the form respectively.

## Related facts

### Corollaries

- Given two subgroups and , their intersection is the subgroup generated by an element with the property that , and if , then . Such an integer is termed a
*least common multiple*of and (if we allow only nonnegative integers, then it is unique). - Given two subgroups and , their join is the subgroup generated by an element with the property that , and if , then . Such an integer is termed a
*greatest common divisor*of and (if we allow only nonnegative integers, then it is unique). - The greatest common divisor of and can be written as for some integers and . That is because it is in the subgroup generated by and .

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Every nontrivial subgroup of the group of integers is cyclic on its smallest element|UP: Introduction four (beginners)|NEXT: No proper nontrivial subgroup implies cyclic of prime order